How Do You Add Velocities in Special Relativity?

Suppose an object A is moving with a velocity v relative to an object
B, and B is moving with a velocity u (in the same direction)
relative to an object C. What is the velocity of A relative to
C?

v
u -------> A
-------> B
C w
----------------->

In non-relativistic mechanics the velocities are simply added and the answer is that
A is moving with a velocity w = u+v relative to C. But in
special relativity the velocities must be combined using the formula

u + v
w = ---------
1 + uv/c2

If u and v are both small compared to the speed of light c,
then the answer is approximately the same as the non-relativistic theory. In the
limit where u is equal to c (because C is a massless particle
moving to the left at the speed of light), the sum gives c. This confirms
that anything going at the speed of light does so in all inertial reference frames.

This change in the velocity addition formula from the non-relativistic to the relativistic theory is not due to
making measurements without taking into account light-travel times, or the Doppler effect. Rather, it is what
is observed after such effects have been accounted for. It is an effect of special relativity which cannot be
accounted for using newtonian mechanics.

The formula can also be applied to velocities in opposite directions by simply changing
signs of velocity values, or by rearranging the formula and solving for v.
In other words, If B is moving with velocity u relative to C
and A is moving with velocity w relative to C then the velocity
of A relative to B is given by

w - u
v = ---------
1 - wu/c2

Notice that the only case with velocities less than or equal to c that is
singular is w = u = c, which gives the indeterminate value zero divided by zero.
In other words, it's meaningless to ask for the relative velocity of two photons that are moving in the
same direction.

How can that be right?

Naively, the relativistic formula for adding velocities might not seem to make sense. But this is due to a
misunderstanding of the idea, which can easily be confused with the following one: suppose the object B
above is an experimenter who has set up a reference frame consisting of a marked ruler with clocks positioned at
measured intervals along it. He has synchronised the clocks carefully by sending light signals along the
line, taking into account the time taken for the signals to travel the measured distances. He now observes
the objects A and C which he sees coming towards him from opposite directions. By watching
the times they pass the clocks at measured distances, he calculates the speeds with which they are moving towards
him. Sure enough, he finds that A is moving at a speed v and C is moving at
speed u. What will B observe as the speed at which the two objects are coming together? It
is not difficult to see that the answer must be u+v whether or not the problem is treated
relativistically. In this situation, the two velocities do add according to ordinary vector
addition.

But that was a different scenario and question to the first one asked above. Originally we asked for the
velocity of C as measured relative to A, and not the speed at which
B observes A and C to approach each other. This is different because the rulers
and clocks set up by B cannot be used to measure distances and times correctly by A, since
for A the clocks do not even show the same time. To go from the reference frame of A to the
reference frame of B, we must apply a Lorentz transformation on co-ordinates in the following way (taking
the x-axis parallel to the direction of travel and the spacetime origins to coincide):

This gives the correct formula for combining parallel velocities in special relativity. A feature of the
velocity addition formula is that if you combine two velocities less than the speed of light, you always get a
result that is still less than the speed of light. This means that no amount of combining velocities can take
you beyond light speed. Sometimes physicists find it more convenient to talk about
the rapidityr, which is defined by the relation

v = c tanh (r/c)

The hyperbolic tangent function tanh maps the real line from minus infinity to plus infinity onto the
interval −1 to +1. So while velocity v can only vary between −c and c,
the rapidity r varies over all real values. At small speeds rapidity and velocity are approximately
equal. If s is also the rapidity corresponding to velocity u, then the rapidity t
of the combined velocities is given by the simple addition

t = r + s

This follows from the identity of hyperbolic tangents

tanh x + tanh y
tanh (x+y) = -------------------
1 + tanh x tanh y

Rapidity is therefore useful when dealing with combined velocities in the same
direction, and also for solving problems with linear acceleration.

For example, if we combine the speed vn times, the result is

w = c tanh [ n tanh-1 (v/c) ]

The velocity addition formula for non-parallel velocities

The previous discussion only concerned itself with the case when both velocities
v and u were aligned along the x-axis; the y and
z directions were ignored.

Consider now a more general case, where B is moving with velocity v =
(vx,0,0) in A's reference frame, and C is moving with
velocity u = (ux, uy, uz) in B's
reference frame. The question is to find the velocity w = (wx,
wy, wz) of C in A's reference frame.
This is still not quite the most general situation, since we are assuming B to be
moving in the direction of A's x-axis, but it is a decent compromise,
since the most general formula is somewhat messy. In any event, one can always
orient A's frame using Euclidean rotations so that B's direction of
motion lies along the x-axis.

There is one additional assumption we will need to make before we can give the
formula. Unlike the case of one spatial dimension, the relative orientations of
B's frame of reference and A's frame of reference is now
important. What B perceives as motion in the x-direction (or
y-direction, or z-direction) may not agree with what A
perceives as motion in the x-direction (etc.), if B is facing in a
different direction from A.

We will thus make the simplifying assumption that B is oriented in the
standard way with respect to A, which means that the spatial co-ordinates of
their respective frames agree in all directions orthogonal to their relative motion.
In other words, we are assuming that

yB = yA
zB = zA

In the technical jargon, we are requiring B's frame of reference to be
obtained from A's frame by a standard Lorentz transformation (also known as a
Lorentz boost).

In practice, this assumption is not a major obstacle, because if B is not
initially oriented in the standard way with respect to A, it can be made to be so
oriented by a purely spatial rotation of axes. However, it should be warned that if
B is oriented in the standard way with respect to A, and C is
oriented in the standard way with respect to B, then it is not necessarily true
that C is oriented in the standard way with respect to A! This
phenomenon is known as precession. It's roughly analogous to the
three-dimensional fact that, if one rotates an object around one horizontal axis and then
about a second horizontal axis, the net effect would be a rotation around an axis which is
not purely horizontal, but which will contain some vertical components.

If B is oriented in the standard way with respect to A, the Lorentz
transformations are given by

xB = γ(vx)( xA - vx tA )
yB = yA
zB = zA
tB = γ(vx)( tA - vx/c2 xA )

Since C is moving along the line

(xB,yB,zB,tB) = (ux t, uy t, uz t, t) (t real),

we see, after some computation, that in A's frame of reference C is
moving along the line

Thus the velocity w = (wx, wy, wz) of
C with respect to A is given by the above three formulae, assuming that
B is orientated in the standard way with respect to A. Note that
if uy=uz=0 then this reduces to the simpler velocity
addition formula given before.

Relative speeds

If an observer A measures two objects B and C to be
travelling at velocities u = (ux, uy, uz) and
v = (vx, vy, vz) respectively, one may ask the
question of what the relative speed between B and C are, or in other
words at what speed wB would measure C to be travelling at, or
vice versa. In galileian relativity the relative speed would be given by

w2 = (u-v).(u-v) = (ux - vx)2 + (uy - vy)2 + (uz - vz)2.

However, in special relativity the relative speed is instead given by the formula